An Elementary Derivation of the Black-hole Area-entropy Relation in Any Dimension

A straightforward two-line derivation of the Bekenstein-Hawking Area-Entropy relation in any dimension is shown based on Shannon’s information theory and Clifford algebras required by the New Relativity Principle. Recently we have proposed that a New Relativity principle may be operating in Nature which could reveal important clues to find the origins of M theory [1]. We were forced to introduce this new Relativity principle, where all dimensions and signatures of spacetime are on the same footing, to find a fully covariant formulation of the p-brane Quantum Mechanical Loop Wave equations. This New Relativity Principle, or the principle of Polydimensional Covariance as has been called by Pezzaglia, has also been crucial in the derivation of Papapetrou’s equations of motion of a spinning particle in curved spaces that was a long standing problem which lasted almost 50 years [6]. A Clifford calculus was used where all the equations were written in terms of Clifford-valued multivector quantities; i.e one had to abandon the use of vectors and tensors and replace them by Clifford-algebra valued quantities, matrices, for example . The String Uncertainty Relations, corrections thereof, and the precise connection between the Regge trajectory behaviour of the string spectrum and the area quantization law of spacetime was also a direct consequence of this New Relativity Principle [2] . Furthermore, The full blown Infinite Dimensional Quantum Spacetime Generalized Uncertainty relations that included the contributions of all p-branes was given [3]. Finally, in [4] we were able to show that there are no such things as EPR paradoxes in this New Scale Relativity ( Machian Relativity) theory in C-spaces [5] which sprang out from Laurent Nottale’s Scale Relativity [7]; Mohammed El Naschie Transfinite Cantorian-Fractal Spacetime [8]; Garnet Ord’s original work on fractal random walks [9] ; the classic Geoffrey Chew’s Bootstrap hypothesis : all p-branes are made of eachother ( William Pezzaglia’s principle of polydimensional covariance ); and p-adic Physics and Non-Archimedean Geometry by Siddarth, Khrennikov, Freund, Volovich, Valdimorov, Zelenov and many others.. There was a one-to-one correspondence between the nested hierarchy of point, loop, 2-loop, 3-loop,......ploop histories encoded in terms of hypermatrices [1] and wave equations written in terms of Clifford-algebra valued multivector quantities.[2-4] This permitted us to recast the QM wave equations associated with the hierarchy of nested p-loop histories, embedded in a target spacetime of D dimensions , where the values of p range from : p = 0, 1, 2, 3......D − 1, as a single QM line functional wave equation whose lines live in a Noncommutative Clifford manifold of 2 dimensions. p = D − 1 is the the maximum value of p that saturates the embedding spacetime dimension. An interacting action line functional, associated with the interacting QFT of lines in Noncommutative Clifford maniolds C-spaces, was launched forward in [2]. The QFT program of such interacting filed theory of C-lines is currently under investigation [10]. Based on this preamble, we are going to present a two line proof of the Bekenstein-Hawking AreaEntropy relation in any dimension. To do so we must introduce some basic definitions : The C-space analog of an invariant ” proper time” is : (dΣ) = (dΩp+1) 2 + Λ(dxdxμ) + Λ (dσdσμν) + Λ (dσdσμνρ) + ....... (1) Λ is the Planck scale in D dimensions : Λ = G 1 D−2 D where GD is Newton’s constant in D dimensions. Polydimensional covariance demands that such scale be the same in every dimension. We set it to unity. In D = 2 one requires to set the Newtonian constant G2 = 1 so that 1 ∞ = 1.